Ab initio density functional theory: OEP-MBPT(2). A new orbital-dependent correlation functional

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 11 15 MARCH 2002

Ab initio density functional theory: OEP-MBPT „2…. A neworbital-dependent correlation functional

Ireneusz Grabowski,a) So Hirata, Stanislav Ivanov, and Rodney J. Bartlettb)

Quantum Theory Project, University of Florida, Gainesville, Florida 32611

~Received 23 July 2001; accepted 29 November 2001!

Using the optimized effective potential~OEP! method in conjunction with an orbital-dependentcorrelation functional developed on the basis of second-order many-body perturbation theory@MBPT~2!#, we performab initio correlated density functional calculations. Unlike other densityfunctional methods, this offers the first correlation approximation in a converging series ofapproximations analogous to that in wave function theory. The OEP-MBPT~2! correlation potentialsof the helium isoelectronic series and the neon atom are in excellent agreement with prior quantumMonte Carlo~QMC! results, while the widely used conventional approximate density functionaltheory correlation potentials have qualitatively incorrect behavior. Furthermore, the correlationenergies, total energies, and the highest occupied orbital energies calculated by the OEP-MBPT~2!method are also remarkably accurate. Unlike QMC, our procedure can be applied to generalmolecules. ©2002 American Institute of Physics.@DOI: 10.1063/1.1445117#

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I. INTRODUCTION

Density functional theory~DFT! based on the Kohn–Sham~KS! scheme1,2 is widely used to study the electronstructure of matter in physics and chemistry. Since the qity of the results is determined by the approximations ufor the exchange-correlation functionals, the developmenmore accurate and at the same time relatively simplepractical approximate functionals is of primary importancThe newest approach for developing exchange-correlafunctionals might be calledab initio DFT,3 since it alleviatesthe primary failing of DFT, namely, that there is no wayguarantee convergence to the right answer. The basic idto take an exchange-correlation energy expression eslished in theab initio wave function theory and to use it aan explicitly orbital-dependent~and hence implicitly density-dependent! exchange-correlation functional in DFT. Thuniversal and parameter-free methodology allows us torectly exploit knowledge from wave function theory and sytematically improve exchange-correlation potentials inKS DFT scheme.

One of the approaches that substantiatesab initio DFT isthe optimized effective potential~OEP! method.4–6 In thisapproach, a given total energy functional depending expitly on single-particle orbitals is minimized under the costraint that these orbitals are the solutions of a single-parKS equation with a local effective potential. Originally, thexchange-only OEP method was derived by TalmanShadwick4 following the idea of Sharp and Horton7 as amethod of giving a variationally optimal local potentiwhose associated single-particle orbitals minimizeHartree–Fock~HF! energy expression. Later in the conteof DFT, the OEP equation was rederived from a differe

a!Permanent address: Institute of Physics, Nicholas Copernicus UniveTorun, Poland.

b!Author to whom correspondence should be addressed.

4410021-9606/2002/116(11)/4415/11/$19.00

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viewpoint by Sham and Schlu¨ter8 and by Go¨rling and Levy.9

The solution of the full integral equation of the OEP methis numerically very demanding and has been achieved soonly for systems with spherical symmetry4,6 or periodicsystems.10,11 In practice, an approximation proposed bKrieger, Li, and Iafrate6,12,13 ~KLI ! is frequently used. Re-cently, the linear-combination-of-atomic-orbital~LCAO! al-gorithms for the OEP method have been developed receby our group14 and by Go¨rling.15 They permit routine OEPcalculations for atoms and molecules with Gaussian-typesis sets, but without any further approximation, and conquently extend the applicability of the OEP method signcantly.

The incorporation of electron correlation into the OEformalism has been the subject of substantial intersince the development of the exchange-only OEP methodmost of the existing applications, the exchange-only Ofunctional has been combined with conventional densfunctionals for the correlation energy,10,16,17 i.e., the localdensity approximation~LDA !, the generalized gradient approximation ~GGA!, or the semiempirical Colle–Salvetfunctional.18,19 It is well known, however, that the successLDA- or GGA-type functionals relies to some extent oncancellation of errors between the exchange and correlawhich is no longer maintained if a highly accurate exchanfunctional such as exchange-only OEP is used. Severalproaches to derive approximate correlation energy functials suitable for the OEP method have been reported inliterature. Go¨rling and Levy9,20 proposed a series of orbitadependent correlation functionals from the coupling-constperturbation expansion, which converges to a formally exrepresentation of the correlation functional. Such a develment has also been studied by Ivanov and Bartlett.21 Casida22

considered including correlation effects in the OEP methwith the aid of the Sham–Schlu¨ter equation and secondorder perturbative approximation to Dyson’s self-ener

ty,

5 © 2002 American Institute of Physics

IP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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4416 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Grabowski et al.

Graboet al.5 offered another scheme to derive a correlatfunctional using a particular perturbative expansion forinteracting Hamiltonian, and derived equations involving tGreen’s functions for the exchange-correlation potentialenergy. Unfortunately, there are no numerical examplespractical applications based on any of the proposed formisms. Recently, however, Engel and co-workers23,24 evalu-ated correlation energies of the Go¨rling–Levy ~GL! pertur-bation theory, although they neglected the single-excitacontribution, and used the converged exchange-only OEPsults. They did not achieve self-consistency betweenexchange-correlation potential and orbitals because theorous solution of the corresponding correlated OEP equawas too complicated.

In this paper, we perform correlated self-consistent Ocalculations for some atoms for which there exist essentiexact correlation energies and potentials.25,26 The OEP cor-relation energy and potential expressions are based onsimplestab initio wave function theory correlation energexpression, namely second-order many-body perturbatheory @MBPT~2!#, and hence we call this new orbitadependent correlation functional treatment OEP-MBPT~2!.In this OEP-MBPT~2! correlated calculations, we solve thOEP integral equation for a given MBPT~2! correlation func-tional within the basis set without any further approximatioand we achieve self-consistency between the potentialorbitals. To test our new correlation functional, we compthe correlation potentials obtained from our MBPT~2! corre-lation functional with the exact correlation potential25,26 andalso with the potentials obtained from widely used local agradient-corrected correlation functionals. We find that,though the total, exchange, or correlation energies calculwith the standard DFT functionals can be reasonable,corresponding correlation potentials show incorrect behavIn contrast, our OEP-MBPT~2! correlation potentials are inqualitatively good agreement in both shape and magnitwith the exact ones, and are a dramatic improvement opreviously developed correlation functionals. As a conquence, our OEP-MBPT~2! correlation functional can perform well for total and correlation energies, for which coventional explicit correlation density functionals work webut also accurately reproduces the correlation potentialsorbital energies, for which the latter functionals fail. Theresults clearly indicate that not only the exchange-correlaenergy but also the potential should be employed as a crion in developing functionals.

II. OEP METHOD AND OEP-MBPT „2… CORRELATIONPOTENTIAL

A. The OEP method

In the KS DFT method,27,28 we express the total energof the system as a sum of the kinetic energy of a systemnoninteracting particles, the Coulombic interaction energythese particles, and interaction energy between the electand the external potentialv(r ) and the exchange-correlatioenergy,


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dl-eder.

eer-

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ne-

offns

E@ra ,rb#5(s

a,b

(i

Ns E dr f is* ~r !S 21

2¹2Df is~r !

11

2E dr dr 8r~r !r~r 8!

ur2r 8u1E dr r~r !v~r !

1Exc@ra ,rb#, ~1!

wherer(r ) is the electron density andNs is the number ofs-spin electrons. The spin densitiesra(r ), rb(r ), and theKS orbitals$fps% are obtained by self-consistently solvinthe KS equation2

$2 12¹

21vss@ra ,rb#~r !%fps~r !5epsfps~r !, s5a,b~2!

where

rs~r !5(i 51

Ns

uf is~r !u2. ~3!

The local effective or KS potentialvss@ra ,rb#(r ) in Eq. ~2!can be written as

vss@ra ,rb#~r !

5v~r !1E dr 8r~r 8!

ur2r 8u1Vxcs@ra ,rb#~r !, ~4!

where

Vxcs@ra ,rb#~r !5dExc@ra ,rb#

drs~r !~5!

is the local exchange-correlation potential formally definas the functional derivative of the exchange-correlationergy with respect to the electron density,rs(r ).

The exchange spin components of the exchancorrelation functional Exc@ra ,rb#, where Exc@ra ,rb#5Ex,a@ra#1Ex,b@rb#1Ec@ra ,rb#, are defined via

(s

a,b

Ex,s@rs#5^F0uVeeuF0&21


ur2r 8u,

~6!

whereVee is the electron-electron interaction operator. In E~6! F0 is the KS single determinant built from theNs , s5a,b, lowest-energy solutions to Eq.~2!. The correlationenergy functionalEc@ra ,rb# is given by

Ec@ra ,rb#

5^Cra ,rbuT1VeeuCra ,rb

&21


ur2r 8u

2(s

a,b H Ex,s@rs#21

2 (i 51

Ns E drf is* ~r !¹2f is~r !J , ~7!

whereT is the kinetic energy operator andCra ,rbis the true

ground-state wave function of the interacting system tyields the ground-state densityr(r ), which is a sum of thespin densitiesra(r ) and rb(r ), i.e., r(r )5ra(r )1rb(r ).By construction the KS single determinantF0 yields rs(r )for s5a,b. Pure-statev representability is assumed.29


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4417J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Ab initio density functional theory

In the OEP method, where we do not restrict ourselveexchange-only OEP of Talman and Shadwick4 but also ‘‘cor-related OEP,’’ the exchange-correlation functional is anplicit, usually approximate, functional of spin orbitaExc

OEP@$fps%# and thus only an implicit functional of the spidensitiesra andrb , in contrast to the conventional explicitly density-dependent exchange-correlation functionals. Tsuperscript OEP indicates that the functionals and potenare now explicitly orbital dependent. The spin orbitals$fps%are the solutions to the single-particle KS equation~2! with alocal effective OEP potentialvss

OEP(r ). This is equivalent torequiring vss

OEP(r ) to be a local potential determined by thcondition that its corresponding orbitals be the ones tminimize the energy functional~1!.4 The stationarity condi-tion

dEOEP@$fpt%#

dvss~r !U

vss5vssOEP

50 ~8!

leads to the following by virtue of the chain rule for funtional derivatives:

(s8

a,b

(pE dr 8H dEOEP@$fqt%#

dfps8~r 8!

dfps8~r 8!

dvss~r !U

vss5vssOEP

1c.c.J50. ~9!

In the above equations and in the following, we use the cvention thati,j,k label occupied orbitals,a,b,c label virtualorbitals, andp,q,r label either. For a given approximation oExc

OEP@$fpt%#, the first factor in the integrand in Eq.~9! iseasily derived. The second factor requires the standard fiorder perturbation theory for an infinitesimal perturbatipotentialdvss ,

dfps8~r 8!

dvss~r !5ds,s8(

qÞp

fps~r !fqs* ~r !

eps2eqsfqs~r 8!. ~10!

Inserting the functional derivative of Eq.~1! and using Eq.~10! into Eq. ~9!, we obtain

(p

(qÞp

H fqs* ~r !fps~r !

eps2eqsE dr 8fps* ~r 8!

3F2 12¹

21E dr 9r~r 9!

ur 82r 9u1v~r 8!1

1

fps* ~r 8!

dExcOEP

dfps~r 8!G3fqs~r 8!1c.c.J 50. ~11!

Using Eq.~2!, we can rewrite the above equation in termsvss

OEP,

(pE dr 8F H vss

OEP~r 8!

2E dr 9r~r 9!

ur 82r 9u2v~r 8!2

1

fps* ~r 8!

dExcOEP

dfps~r 8!J3 (

qÞp

fqs* ~r !fqs~r 8!

eps2eqsfps* ~r 8!fps~r !1c.c.G50, ~12!


to

-

els

t

-

st-

f

and identifying

VxcsOEP~r !5vss

OEP~r !2E dr 8r~r 8!

ur2r 8u2v~r !, ~13!

we can rewrite Eq.~12! in the standard form of the OEPequation

(pE dr 8Ffps* (r 8)H Vxcs

OEP(r 8)

21

fps* ~r 8!

dExcOEP

dfps~r 8!J Gps~r ,r 8!fps~r !1c.c.G50, ~14!

where

Gps~r ,r 8!5 (qÞp

fqs~r !fqs* ~r 8!

eps2eqs. ~15!

The OEP integral equation~14! has to be solved for theexchange-correlation potentialVxc

OEP in each KS~OEP! itera-tion, and the self-consistency, Eq.~2!, must be achieved athe end.

By rewriting Eq. ~14!, we can formally represent thexchange-correlation potential as

VxcsOEP~r !

5(pE H dExc

OEP@$fqt%#

dfps~r 8!

1

fps* ~r 8!1c.c.J Xss

21~r 8,r !dr 8,

~16!

where Xss21(r 8,r ) is the inverse of the static KS linear re

sponse function of a system of noninteracting particles

Xss~r ,r 8!5drs~r !

dvss~r 8!

5(i

Ns

(a

f is* ~r !fas~r !fas* ~r 8!f is~r 8!

e is2eas1c.c.

~17!

It should be understood thatXss21(r ,r 8) is defined in the do-

main of the eigenfunctions ofXss(r ,r 8) and singularities inXss(r ,r 8) do not pose any formal problem in Eq.~16!.30 Inthe LCAO-OEP procedure, the potential and the respofunction ~17! are represented in a real atomic orbital~AO!basis$gp(r )% as

VxcsOEP~r !5(

pcpgp~r ! ~18!

and

Xss~r ,r 8!5(p,q

~Xss!pqgp~r !gq~r 8!, ~19!

where

~Xss!pq5(i ,a

S ~ i sasup!~ i sasuq!*

e is2eas1c.c.D ~20!

and (r sssup) is the overlap integral of the orbitalsf rs(r ),fss(r ), and the orthonormal auxiliary Gaussian functiogp(r ),


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~r sssup!5E f rs* ~r !fss~r !gp~r !dr . ~21!

Hence,Xss21(r ,r 8) can be expanded by the same basis as

Xss21~r ,r 8!5(

p,q~Xss

21!pqgps* ~r !gqs~r 8!, ~22!

where the inverse matrixXss21 can be obtained by singula

value decomposition ofXss .The exchange-correlation energy functional can be p

titioned into the exchange and correlation components,Exc

OEP@$fpt%#5ExOEP@$fpt%#1Ec

OEP@$fpt%#, and likewise theexchange and correlation potentials can be separated, aOEP equation is a linear integral equation. Choosing theexchange energy functional in terms of KS orbitals asexchange functionalEx

OEP,

ExOEP@$fpt%#52

1

2 (s

a,b

(i , j

Ns

~ i s j su j si s!, ~23!

we obtain the exchange potential in the following form:3,14

VxsOEP~r !52(

p,q(i , j ,a

F H ~ i s j su j sas!

e i s2eas

~asi suq!J3~Xss

21!qpgp~r !1c.c.G , ~24!

where thechemical notation for two-electron integrals iused:

~psqsur ssk!

5E fps* ~r !fqs~r !1

ur2r 8uf rs* ~r 8!fss~r 8!dr dr 8. ~25!

B. The orbital-dependent correlation functional andpotential

In this study, we shall employ an explicitly orbitadependent correlation energy functional and potential onbasis of the second-order Rayleigh–Schro¨dinger~RS! pertur-bation theory expression. We start with the Hamiltonian osystem of interacting electrons in the standard normal oform:31

H5(p,q

f pq$p†q%11

4 (p,q,r ,s

^pquurs&$p†q†sr%

1^F0uHuF0& ~26!

5(p

f pp$p†p%1 (p,q

pÞq

f pq$p†q%1W1^F0uHuF0&,

~27!

where $p†q% and $p†q†sr% denotes the normal product ospin orbital second quantized operators and the last t^F0uHuF0& is the expectation value of the Hamiltonianthe KS determinantF0 , which is the single~Slater! determi-nant built from theN5Na1Nb lowest-energy spin orbitalswhereN is the total number of electrons in system of inte


r-.,

theFe

e

aer

m

est. The fully contracted part of the Hamiltonian is the epectation value. In the second term we use following notion for two electron integrals:

^pquurs&5~pruqs!2~psuqr !. ~28!

In Eq. ~26! we have introduced the usual Fock matrix ements defined in terms of KS-OEP spin orbitals:

f pq5~puhuq!1^pu(jE f j* ~2!

31

r 12~12P12!f j~2!dt2q

5~puhuq!1(j

$~pqu j j !2~p j u jq !%, ~29!

where (puhuq) is the one-electron integral

~puhuq!5E fp* ~r !h~r !fq~r !dr , ~30!

andh is thecore Hamiltonian~for a given electron! describ-ing its kinetic energy and potential energy in the field of tnuclei.

In this study we choose the zero-order HamiltonianH0

for the perturbation expansion as follows:

H05(p

ep$p†p%. ~31!

From the usual RS perturbation theory, we can writesecond-order many-body correlation energy express@MBPT~2!# in the following form:

Ec~2!5^F0u~Vee2Vuxc!R0~Vee2Vuxc!uF0&, ~32!

@Vuxc5( i 51Vuxc( i )# but with the understanding that

Vuxc5lVx1lVc~2!1lE dr 8

r~r 8!

ur2r 8u, ~33!

which makes all other terms in Eq.~27! part of the perturba-tion, ordered byl. It should be apparent that since the corelation potentialVc

(2) is formally the functional derivative ofEc

(2) , it has to arise in second and higher orders of MBPwhile the exchange potential is first order. We temporarintroduce the ‘‘order parameter’’l to make this explicit.Hence, the energy functional that is only second-order ilsuppresses theVc

(2) , meaning it is exactly the same functional as that in Go¨rling-Levy perturbation theory. This distinction will be important below. The resolvent

R05~E02H0!21Q, ~34!

whereQ is the projector for the orthogonal complementF0 ~i.e., single and double excitations!, and E0

5^F0uH0uF0&. Henceforth, we use the simplified notatioVx andVc

(2) for the sum ofa- andb-spin parts of the localeffective exchange and correlation potentials, respectivThe sum of thea andb parts of the left-hand side of Eq.~16!is therefore identified asVx1Vc

(2) . We can write this expression in a more explicit well-known spin orbital form:32,33


ns

reled

n

perzetoain,

dal

nlyhichtch-

b-


Ec~2!5EDc

~2!1ESc~2!

51

2 (i , j ,a,b

u~ iau jb !u2

e i1e j2ea2eb2

1

2 (i , j ,a,b

~ iau jb !~a j ubi !

e i1e j2ea2eb

1(i ,a

u f iau2

e i2ea, ~35!

where EDc(2) and ESc

(2) are the double- and single-excitatioparts of the MBPT~2! correlation energy functional. In thiexpression, it should be understood thatf pq5epdpq

2l@Sk(pkukp)1^puVxup&#2l2^puVc(2)up&. Hence, limiting

ourselves to consistent orders of perturbation theory, theno contribution from the correlation potential to the singexcitation,f ia term. Instead, we have the same second-orexpression as that in the GL perturbation theory, plusproper linear equation forVc as required by Eq.~16!. Fur-thermore, we ensure that the KS density does not cha

nth

ra


is-era

ge

from its self-consistent, converged value. In a future pathis point will be made explicitly, where we also generalithe theory to infinite order in perturbation theory andcoupled-cluster theory. The second-order solution we obtalso provides the high-density limit in DFT.

The correspondings-spin correlation potential, obtaineby taking the derivative of the correlation energy function~35! with respect to densityVcs

(2)(r )5dEc(2)/drs(r ), can be

readily derived. For the sake of compactness, we opresent the expressions for the spin-restricted case, wcan be readily implemented for closed-shell systems. Swiing to the spatial orbital notation~now i, j, k, l denote occu-pied spatial orbitals,a, b, c, dvirtual spatial orbitals, andseither, andp, q denote orthonormal auxiliary functions! anddropping the spin indices to simplify the expression, we otain

Vcs~2!~r !5VDcs

~2! ~r !1VScs~2! ~r !

5(pq

H (i jab

2~ iau jb !2~a j ubi !

e i1e j2ea2ebF2(

sÞ i

~sau jb !

e i2es~ isuq!12(

sÞa

~ isu jb !

ea2es~sauq!2

1

2

~ iau jb !

e i1e j2ea2eb@~ i i uq!1~ j j uq!

2~aauq!2~bbuq!#G J ~Xss21!pqgp~r !1(

pqH(

ia

f ia

e i2eaF2(

sÞ i

~siuq!

e i2esf sa12(

sÞa

~sauq!

ea2esf is

12(kc

~ckuq!

ek2ec@4~ iauck!2~ icuka!2~ ikuca!#2

f ia

e i2ea@~ i i uq!2~aauq!#G J ~Xss

21!pqgp~r !, ~36!

whereVDcs(2) (r )5dEDc

(2)/drs(r ) andVScs(2) (r )5dESc

(2)/drs(r ).We can rewrite Eq.~36! in a more explicit way, suppressing the superscript,

Vcs~2!~r !5VDcs8 ~r !1VDcs9 ~r !1VScs8 ~r !1VScs9 ~r !, ~37!

where

VDcs8 ~r !5(pq

H (i jab


e i1e j2ea2ebF2(

c

~cau jb !

e i2ec~ icuq!12(

k

~ iku jb !

ea2ek~kauq!G J ~Xss

21!pqgp~r !, ~38!

VDcs9 ~r !5(pq

H (i jab


e i1e j2ea2ebF2(

lÞ i

~ lau jb !

e i2e l~ i l uq!12(

dÞa

~ idu jb !

ea2ed~dauq!2

1

2

~ iau jb !

e i1e j2ea2eb

3@~ i i uq!1~ i j uq!2~aauq!2~bbuq!#G J ~Xss21!pqgp~r !, ~39!

VScs8 ~r !5(pq

H(ia

f ia

e i2eaF2(

c

~ciuq!

e i2ecf ca12(

k

~kauq!

ea2ekf ik12(

kc

~ckuq!

ek2ec@4~ iauck!2~ icuka!2~ ikuca!#G J

3~Xss21!pqgp~r !. ~40!

VScs9 ~r !5(pq

H(ia

f ia

e i2eaF2(

j Þ i

~ j i uq!

e i2e jf ja12(

bÞa

~bauq!

ea2ebf ib2

f ia

e i2ea@~ i i uq!2~aauq!#G J ~Xss

21!pqgp~r !, ~41!

ma-

rvery

where we separated termsVDcs8 (r ) defined by Eq.~38! andVScs8 (r ) defined by Eq.~40! from the rest, as they maintaithe same structure of integral equation for potentials asexchange-only OEP integral equation~24!. The termsVDcs9 (r ) defined by Eq.~39! andVScs9 (r ) defined by Eq.~41!result in a more complicated structure for the OEP integequation.

e

l

From the above expressions~38!–~41! for the correlationpotential, we can immediately see some possible approxitions, which will simplify our OEP-MBPT~2! correlationtreatment. First of all, we can neglect terms~40! and ~41!,which come from the singly excited part of the MBPT~2!energy functional~35!. This should not seriously affect oucalculated results, because these terms should give a


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small amount to the correlation energy and correlation potial, since the HF and OEP occupied orbitals are similar.will call this approximation OEP-MBPT~2!D.

Another possible approximation is obtained by negleing terms ~39! and ~41!, which have a different and morcomplicated structure than the exchange-only OEP ingral equation. We will label it by OEP-MBPT(2)S8D8.We can also use these two types of approximationsthe same time, and obtain the simplest expression forcorrelation potential, which is Eq.~38! only. We will call itOEP-MBPT(2)D8. The correlated OEP method that contaiall terms~38!–~41! will be called OEP-MBPT~2!SD or justOEP-MBPT~2!.

In each self-consistent field cycle, we first evaluatematrix representation of the response functionXss ~17! andthen we invert this matrix by the singular value decompotion procedure to ensure that any singularity is excludfrom theXss

21 evaluation. We then evaluate matrix elemenof the exchange-correlation potential over atomic orbit$xm%, i.e., ^muVxc

OEPun&, using Eqs.~24! and ~36!, and pro-ceed to a self-consistent solution that includes theVc

(2) po-tential. For numerical details applied toVx see Refs. 14 and30.

III. RESULTS AND DISCUSSION

A. The helium atom and the helium isoelectronicseries

To assess the performance of our correlation functiowe will first concentrate on the helium isoelectronic seribecause there exist practically exact solutions25 of the Schro¨-dinger equation for these systems,25,34and they are very useful for benchmark comparisons of various quantities otained from different~DFT and OEP in our case! methods.For two-electron systems including the helium isoelectroseries, the exact local exchange potential is also knowhich makes the discussion of pure correlation effestraightforward.

We start by comparing the results for the helium atoThe OEP-MBPT~2! calculations~full version and differentapproximations! were performed using the basis set whiconsisted of even-tempered 20s-type, 10p-type and 2d-typefunctions.35 We compare them with the DFT results calclated with the conventional exchange-correlation functionBLYP is the Becke gradient corrected exchange function36

~Becke88! and Lee–Yang–Parr correlation functiona37

~LYP!, and SVWN is the local density approximation to thexchange functional38 ~LDA ! and Vosko–Wilk–Nusair cor-relation functional39 ~VWN!. In our comparison, we also include the results obtained from OEP-KLICS,19 which is theexchange-only OEP method in conjunction with the semialytical approximation due to Krieger, Li, and Iafrate6 ~KLI !,plus a correlation contribution from the Colle anSalvetti18,40 ~CS! correlation functional.

In Fig. 1, we plot the exchange-correlation potential otained from these approximate functionals and the OMBPT~2!SD method for the helium atom. Since the potetials obtained from a LCAO OEP-MBPT~2! procedure areexpanded by Gaussian functions, which do not have the


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rect (21/r ) asymptotic behavior of the potential for larger,although formally correct, the potentials decay too rapidlyvery large r. Consequently, the potential obtained fromLCAO OEP-MBPT~2! procedure is displaced from the trusolution of Eq.~14!, although the shape of the potentialaccurately reproduced in the physically important region.this study, we shift the correlation potential and the magtude of the shift is determined by a highest occupied molelar orbital ~HOMO! condition, analogous to that foexchange-only OEP,6 for a correlated OEP-MBPT~2! calcu-lation. For the energy functional~35!, we can write this con-dition as follows:

E fHs~r !VxcsOEPfHs~r !dr

52(j

~Hs j su j sHs!1SHsHs~2! ~eHs!, ~42!

wherefHs(r ) is the highest occupied orbital. The first teron the right-hand side in Eq.~42! is the expectation value othe nonlocal HF exchange potential constructed for the hiest occupied KS orbitals, andSpq

(2)(E) is a second-order selfenergy expression32

Spq~2!~E!5 (

j ,a,b

~apub j !$2~qau jb !2~ jauqb!%

E1e j2ea2eb

2 (i , j ,b

~ ipu jb !$2~qiub j !2~biuq j !%

e i1e j2E2eb, ~43!

where the summation over spatial orbitals is involved. Torbital energies presented in Tables III and IV are obtainafter the exchange-correlation potentials of the LCAO OEMBPT~2! are corrected for the displacement. Also the OEMBPT~2! exchange-correlation and correlation potentipresented in Figs 1, 2, 3, 5 and 6 are shifted, so that

FIG. 1. Comparison of the exchange-correlation potentials of the helatom obtained from the OEP-MBPT~2!, exchange-only OEP, SVWN, andBLYP functionals.


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HOMO condition is satisfied. The constant shift for the crelation potential has a very small value in comparison toshift calculated for the exchange potential. For example,the neon atom for which numerical results are presenteSec. III B, those shifts are 0.0042 and 0.1570 a.u. for colation and exchange potentials, respectively.

The total shape and magnitude of the OEP-MBPT~2!potential are almost the same as those of the exchangeOEP potential. This is expected because the correlationtential is a small fraction of the exchange-correlation oand correlation hardly affects the overall shape and behaof the total exchange-correlation potential. However, aswell recognized this small correction is essential in descing chemistry correctly. Note that the OEP-MBPT~2!exchange-correlation potential has the correct21/rasymptotic behavior forr ,4.7 after the correction based othe HOMO condition is made, whereas the SVWN aBLYP potentials do not. The small oscillations of the OEMBPT~2! potentials near the nuclei and the deviations frothe 21/r curves at larger (r .4.7) are numerical basis-seeffects in the LCAO-OEP method, and are discussed in dein Ref. 30.

In Table I, we show the total ground-state and exchancorrelation energy and also the exchange and correlacontributions to the total energy calculated from differeDFT and OEP calculations in comparison with the exvalues.25 Despite the fact that the SVWN and BLYP potetials are substantially different from the exact potential,values of the energy components are reasonable; in particthe BLYP functional provides accurate exchange and colation energies. Comparing the results with the exact valuit is obvious that the OEP-MBPT~2! and KLICS schemesperform much better than the conventional KS approacand give qualitatively similar results. Comparing differeapproximate variants of the OEP-MBPT~2! methods, we cansee that there is almost no significant differences in toenergies among them. A larger difference in total energbetween the OEP-MBPT~2!SD and OEP-MBPT~2!D meth-ods come from different expressions for the energy futional, i.e., an additional~negative! singly excited termsESc

@Eq. ~35!#. In Fig. 2, we compare the exact correlation ptential with the OEP-MBPT~2!D8, KLICS, and conventiona

TABLE I. Comparison of the total ground-state energies and exchangecorrelation energies~in hartrees! for the helium atom obtained from differenconventional DFT and OEP calculations. DFT~SVWN, BLYP! and OEP-MBPT~2! calculations were performed using an even-tempered 20s10p2dbasis set.

Method Etot Exc Ex Ec

SVWN 22.834 835 20.973 320 20.861 853 20.111 468BLYP 22.907 066 21.062 048 21.018 344 20.043 704

KLICSa 22.903 3 21.069 1 21.027 5 20.041 6OEP-MBPT~2!D 22.907 800 21.069 036 21.022 800 20.046 236

OEP-MBPT~2!SD 22.907 923 21.069 159 21.022 800 20.046 359OEP-MBPT(2)D8 22.907 773 21.071 437 21.025 306 20.046 131

OEP-MBPT(2)S8D8 22.907 812 21.071 477 21.025 306 20.046 171Exactb 22.903 724 21.066 676 21.024 568 20.042 107

aReference 19.bReference 26.


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-

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DFT correlation potentials, i.e., VWN and LYP. It is evidethat, except for the OEP-MBPT~2! correlation potentials, allthe approximate correlation potentials show very large detions from the exact one. In the region with large electrdensity, these approximate correlation potentials~VWN,LYP, CS! have the wrong sign and qualitatively incorreshapes. In contrast, our OEP-MBPT~2! correlation potentialsare in qualitatively good agreement in shape and magnitwith the exact one. There is a slight difference betweenOEP-MBPT~2!D and OEP-MBPT(2)D8 potentials, the vari-ous approximations to the OEP-MBPT~2! method do not ingeneral cause any qualitative change in the correlationtentials.

In Table II, we compare the total ground-state energfor the helium isoelectronic series obtained from SVWBLYP, KLICS, exchange-only OEP, and OEP-MBPT~2! po-tentials with the exact results.34 The calculations for He, Li1,Be21, and B31 were performed with an even-tempere20s10p2d basis set, and for the other ions we used ancontracted ROOS-ATZP basis set.44 In the last row, the av-erage absolute deviations from the exact values aresented. As we can see from the table, the KLICS and OMBPT~2! total energies are much closer to the exact valthan those obtained from the SVWN or BLYP calculationThe average absolute deviation from the exact values ithe same order for both the exchange-only OEP and OMBPT~2! methods, while KLICS results are slightly betteHowever, for the highly charged two-electron ions, the quity of the KLICS result is worse, while the quality of thOEP-MBPT~2! results becomes better. For Mg101 onwards,the OEP-MBPT~2! total energies are closer to the exact rsults than the KLICS ones. We would like to mention theare almost no differences in total energies between

nd

FIG. 2. Comparison of the exact correlation potential of the helium atwith the correlation potentials obtained from the OEP-MBPT~2!, KLICS,SVWN, and BLYP functionals. The exact correlation potential from Ref. 2



TABLE II. Total ground-state energies~in hartrees, sign reversed! for the helium isoelectronic series from various conventional DFT and OEP-MBPT~2!Dcalculations. In parenthesis are given the correlation energies obtained from OEP-MBPT~2!D calculations. Basis sets used in DFT~BLYP, SVWN! andOEP-MBPT~2!D calculations are described in the text.

BLYP SVWN KLICSa OEP-MBPT~2!D Exactb

He 2.907 066 2.834 835 2.9033 2.907 800~0.046 236! 2.9037Li1 7.279 429 7.142 818 7.2803 7.281 012~0.044 628! 7.2799Be21 13.650 025 13.444 694 13.6556 13.654 044~0.043 108! 13.6556B31 22.019 407 21.742 617 22.0301 22.028 263~0.042 047! 22.0310C41 32.388 272 32.038 680 32.4045 32.402 130~0.041 000! 32.4062N51 44.758 082 44.333 188 44.7788 44.777 063~0.040 983! 44.7814O61 59.127 244 58.626 541 59.1531 59.151 432~0.040 967! 59.1566F71 75.496 340 74.918 991 75.5274 75.526 898~0.040 917! 75.5317Ne81 93.865 423 93.210 726 93.9017 93.901 800~0.040 855! 93.9068Na91 114.236 751 113.504 042 114.2761 114.276 651~0.040 894! 114.2819Mg101 136.606 019 136.795 111 136.6505 136.651 638~0.040 909! 136.6569Al111 160.975 412 160.085 744 161.0250 161.026 592~0.040 897! 161.0320Si121 187.344 860 186.376 105 187.3995 187.401 562~0.040 894! 187.4070Ar161 312.822 695 311.535 012 312.8997 312.901 355~0.040 861! 312.9072

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OEP-MBPT~2!D and OEP-MBPT(2)D8 methods. Also add-ing the single excitations in OEP-MBPT~2!SD andOEP-MBPT(2)S8D8 do not change the total energies.

To assess the quality of the exchange-correlation potials, it is informative to look at the highest occupied orbienergies. In the limit of an exact exchange-correlation futional, this value should be equal to the negative of the iization potential of the system.11,41–43These values are versensitive to the quality of the exchange-correlation potenIn Table III, we present the highest occupied orbital energobtained from various approximate functionals. It is obviothat, once the HOMO condition has been imposed, the OMBPT~2! method has to give very good results. This suprior quality is due to the inclusion of the exact exchanthrough the OEP scheme, leading to the correct asymp


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behavior in the KS potential, and it is further improved bthe OEP-MBPT~2! correlation treatment. In the KLICS results, one might expect similar quality, because the Kexchange-only potentials are good approximations toexchange-only OEP. However, as we can see, addingColle–Salvetti correlation potentials to the exchange-oKLI potentials deteriorates the results. This is expected sithe CS correlation potentials have the wrong sign inphysically relevant regions of space and as has already bconfirmed by the plot of the correlation potentials in Fig.As in the total ground-state energy case, the highest occuorbital energies calculated by the OEP-MBPT~2!D, OEP-MBPT~2!SD, OEP-MBPT(2)D8, and OEP-MBPT(2)S8D8methods are almost the same.

nd

7

TABLE III. The highest occupied orbital energies~in hartrees, signs reversed! for the helium isoelectronic series from various conventional DFT aOEP-MBPT~2!D calculations. Basis sets used in DFT~BLYP, SVWN! and OEP-MBPT~2!D calculations are described in the text.

BLYP SVWNKLI

x onlybOEP

x only KLICSb OEP-MBPT~2!D Exactb

He 0.5894 0.5704 0.9180 0.9180 0.9446 0.8904 0.9037Li1 2.2312 2.1903 2.7924 2.7924 2.8227 2.7719 2.7799Be21 4.8760 4.8066 5.6671 5.6671 5.6992 5.6482 5.6556B31 8.5199 8.4206 9.5420 9.5418 9.5751 9.5235 9.5310C41 13.1634 13.0335 14.4169 14.4166 14.4507 14.3983 14.4062N51 18.8067 18.6455 20.2918 20.2914 20.3261 20.2738 20.2814O61 25.4498 25.2567 27.1668 27.1662 27.2014 27.1492 27.1566F71 33.0928 32.8676 35.0418 35.0410 35.0766 35.0245 35.0317Ne81 41.7357 41.4780 43.9167 43.9158 43.9517 43.8997 43.9068Na91 51.3794 51.0890 53.7917 53.7915 53.8269 53.7749 53.7819Mg101 62.0223 61.6990 64.6667 64.6665 64.7020 64.6500 64.6569Al111 73.6652 73.3088 76.5417 76.5415 76.5770 76.5251 76.5320Si121 86.3081 85.9185 89.4167 89.4165 89.4521 89.4002 89.4071Ar161 146.8793 146.3556 150.9166 150.9164 150.9523 150.9005 150.902

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In Fig. 3, we plot the OEP-MBPT(2)D8 correlation po-tentials of some selected atoms in the helium isoelectroseries. The shapes and magnitudes of the plotted correlapotentials are in good agreement with the exact resultssented in Fig. 4 of Ref. 25~note that the horizontal axes oFig. 3 are not scaled by the nuclear chargeZ, unlike in Fig. 4of Ref. 25!. As we can see in the figure, the position of tmaximum of the correlation potential is shifted toward tnucleus for heavier ions. This can be simply explained bydependence of the correlation potential on the shell strucof the electron density. In Fig. 4, we plot the radial chardensity obtained from the OEP-MBPT~2! calculations for afew members of the helium isoelectronic series. The potions of maxima in both figures match very well.

B. Neon atom

As another example, we have performed OEP-MBPT~2!calculations for the neon atom in an uncontracted ROOATZP basis set. In Table IV, we compare the total grounstate energies and highest occupied orbital energies obtafrom OEP-MBPT~2!, KLICS, SVWN, and BLYP calcula-tions with the exact results.45 Again the correlated OEP caculations give much better total energies and highest ocpied orbital energies than the conventional DFT method. Tapproximate KLICS total energies are slightly closer toexact result than the total energy of the OEP-MBPT~2!method, although the latter result is already highly accurThe total energies calculated in the OEP-MBPT~2! methodsare too negative, because the GL expression for the secorder energy functional that we use in our calculationsready gives much lower energies than Hartree-Fock baMBPT~2!, e.g. The highest occupied orbital energies otained from the OEP-MBPT~2! calculations are also o

FIG. 3. Correlation potentials of selected members of the helium isoetronic series obtained from the OEP-MBPT(2)D8 calculations.


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slightly poorer quality than those of KLICS. Again, like ithe case of the helium atom, adding the CS correlationtential worsens the orbital energy relative to the Kexchange-only result, again indicating that the CS correlapotential has the wrong sign in the physically relevant regjust as for the helium atom. In the OEP-MBPT~2! case, thecorrelation correction moves in the correct direction becaof the qualitatively correct correlation potential~see Fig. 6!,but the magnitude of the correction is slightly too large.Fig. 5, we plot the exchange-correlation potential obtainfrom the OEP-MBPT~2! calculations and the exchange ptential from the exchange-only OEP calculations. As epected, the exchange contribution largely determinesshape of the total exchange-correlation potential, althothe effect of correlation is visible near the intershell regiand is quantitatively very important. In Fig. 6, we plot thLYP, VWN, and OEP-MBPT~2! correlation potentials. We

c-FIG. 4. Radial density of the selected members of the helium isoelectrseries obtained from the OEP-MBPT(2)D8 calculations.

TABLE IV. Total ground-state energies and the highest occupied orbenergieseHOMO ~in hartrees, signs reversed! for the neon atom. DFT~BLYP,SVWN! and OEP-MBPT~2! were performed using an uncontracted ROOATZP basis set.

Etot eHOMO

BLYP 128.972 478 0.4914SVWN 128.232 975 0.4980KLI x only 128.545 4 0.8494KLICSa 128.920 2 0.8841OEP-MBPT~2!D 128.987 017 0.6490OEP-MBPT~2!SD 129.009 088 0.6489OEP-MBPT(2)D8 128.985 505 0.6546OEP-MBPT(2)S8D8 128.995 330 0.6545Exactb 128.937 6 0.7945



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compare them with the exact correlation potential26 calcu-lated from very accurate charge density obtained from qutum Monte Carlo calculations~QMC!. The conventionalDFT correlation potentials have the completely wrong shas compared to the exact~QMC! one. Our OEP-MBPT~2!correlation potential shows qualitatively the correct sha

FIG. 5. Comparison of the exchange-correlation potential with exchapotential of the neon atom obtained from the OEP-MBPT~2! and exchange-only OEP calculations, respectively.

FIG. 6. Comparison of the exact~QMC! correlation potential of the neonatom with the correlation potentials obtained from the OEP-MBPT~2!,SVWN, and BLYP functionals. The exact~QMC! correlation potential fromRef. 26.


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although the magnitude of the OEP-MBPT~2! correlation po-tential is almost twice as great as the exact~QMC! one.Overall the OEP-MBPT~2! correlation potential is a strikingimprovement over those obtained from popular functionaIt may also be noticed that the shape of the correlationtentials reflects the shape of the corresponding radial chdensity. This is analogous to the well-known observation tthe humps in the exchange potentials occur owing tointershell structure of the radial charge density. One miexpect that a correlation potential will reflect the radcharge density for an atom and allow the electrons to aveach other. This simple argument appears to be supporteFigs. 2 and 3 for He and by Figs. 6 and 7 for Ne, wherecan find that the peaks in the correlation potentials coincwith those in the radial charge densities. A comparison ofradial charge density obtained with accurate orbitdependent correlation potentials should compare well wthose fromab initio correlated methods like coupled-clusttheory.

IV. CONCLUSIONS

We have performed anab initio correlated DFT calcula-tion employing the OEP-MBPT~2! correlation functional andobtained accurate exchange-correlation and correlationtentials for some atoms whose exact exchange and cortion potentials are known. We have performed fully seconsistent OEP-MBPT~2! calculations to achieve selfconsistency between the exchange-correlation potentialorbitals. The total exchange-correlation potentials, whplotted, are hardly affected by electron correlation asexchange potentials dominate in the total exchancorrelation potentials. This is expected as the correlationfects are subtler and are at least an order of magnit

eFIG. 7. Radial density of the neon atom obtained from tOEP-MPBT(2)D8 calculations.


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smaller than the exchange. However, the correlation effevery important in obtaining accurate total energies andbital energies, and the OEP-MBPT~2! treatment systematically improves the exchange-only OEP results across difent properties. These OEP-based methods—exchangeOEP and OEP-MBPT~2!—give significantly better resultsthan conventional DFT functionals such as SVWN or BLYFor correlation potentials, only our OEP-MBPT~2! calcula-tions among the other functionals examined in this studyreproduce the correct shape of the correlation potential,simultaneously give highly accurate results for other quaties such as the total and orbital energies. Additionallyfound almost no difference between numerical resultstained from different approximations of the OEP-MBPT~2!methods, though observable in the potential itself. The acrate prediction of potentials and orbital energies achievedOEP-MBPT~2! is essential in, for example, the calculatioof ionization potentials and excitation energies by timdependent DFT. Our findings also illustrate that not onlyexchange-correlation energy, but also the correspondingtential should be employed as a criterion in developing nfunctionals. We emphasize that our OEP-MBPT~2! correla-tion potentials are perhaps currently the most accurate othat can be obtained for general atoms and molecules agwhich other approximate functionals should be testedcalibrated, and will be instrumental in developing more acurate and systematically improvable exchange-correlafunctionals. The natural extension to the higher ordersperturbation theory and to infinite-order coupled-clustheory should provide a series of converging correlationtentials, analogous to the well known convergingab initiocorrelated approximations, MBPT(2),CCD,CCSD;MBPT(4),CCSD~T!,CCSDT,¯,full CI.

ACKNOWLEDGMENTS

This work was supported by the U.S. Air Force OfficeScientific Research under Grant No. F49620-98-0116 andNational Science Foundation under Grant No. 990015.thank Mark Casida for interesting discussions.

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Ab initio density functional theory: OEP-MBPT(2). A new orbital-dependent correlation functional

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Transcript of Ab initio density functional theory: OEP-MBPT(2). A new orbital-dependent correlation functional